Effect of Couette component on the stability of Poiseuille flow of a Bingham fluid–porous system: Modal and non-modal approaches

In this study, both modal and non-modal stability analyses are attempted in case of Couette–Poiseuille flow of a Bingham fluid overlying a porous layer. Such a flow configuration is widely encountered in the geophysical context in case of oil drilling. The solution of the modal problem yields no unstable eigenvalue, similar to the flow of a viscoplastic fluid in a non-porous channel configuration. Thus, non-modal analysis is performed to throw light on the short-time characteristics. The primary goal is to unveil the complex interplay between the upper plate velocity (Couette component) and the parameters characterizing the porous layer in dictating the flow transition characteristics. The current study is possibly the first attempt at investigating the effect of the Couette flow on the stability of a fluid–porous system for any kind of non-Newtonian fluid and reveals marked departure from the results reported in the literature for a similar flow configuration involving Newtonian rheology. The reason for the deviation is attributed to the role of yield stress, quantified by the Bingham number, and its complex interaction with the Couette number and porous layer parameters (depth, permeability, anisotropy, inhomogeneity, etc.). The relative interaction between fluid and porous modes in an environment of non-linear viscosity variation (owing to the rheology of the viscoplastic fluid), coupled with enhanced shearing (imparted by the Couette component), is found to demonstrate unique, non-monotonic flow transition characteristics. The possible physical mechanism governing short-time (non-modal) amplifications via interaction between the mean shear flow and the perturbation waves is also explored in detail.